Here is what you need to know about [un]employment statistics.
The statistics are managed by the federal Bureau of Labor Statistics (
http://www.bls.gov). Every month they report the previous month's stats, including the employment
level, the unemployment
rate, and so forth. The statistics are measured different from each other, using two different surveys.
This URL always gives the current summary report.
The current employment level, as reported in how many millions of people are currently working at
payroll jobs, is gathered by surveying 150,000 businesses and government agencies, and then projecting the data onto the industries they represent. IE, they call up a bunch of construction companies every month, ask how many construction workers ARE ON THE PAYROLL and then project that answer onto the rest of the industry to guesstimate the current number of PAYROLL construction workers. If this number goes up in a category or the net number goes up then it means that an industry or the economy as a whole has created more jobs than it has shed, and if a number goes down then it means the opposite.
Meanwhile, the official UNEMPLOYMENT RATE is measured by calling random households and asking if anybody in the household is currently unemployed
and looking for a job. They do not count retirees, housewives, students, the self-employed, and other persons who are not "employed" but are not actively looking for a job. The BLS also includes stats on the "long-term" unemployed, but those numbers are not removed from the current unemployed, and instead are reported twice. The BLS also includes stats on the number of people who have "stopped looking" and decided to become full-time moms or go back to school or whatever.
Since one number measures industry payroll, and the other number measures people who are unemployed and looking for work, the numbers are often disconnected from each other. For example (hypothetical), it is possible for a million payroll employees to suddenly quit their jobs and go back to college, resulting in lower employment (less payroll employees) and lower unemployment (less people "looking" for jobs) both at the same time.
There is another "unofficial" employment number which is gathered from the household surveys, which shows that actual employment as measured by household surveys is always much higher than the payroll numbers. This number captures people who do things like make a living on eBay, or run a catering business out of their home, or work construction as sub-contractor not on payroll, and who are not captured by the institutional payroll surveys.
Unemployment insurance is separate from all of that. It is funded and managed by each of the individual states (although Congress frequently passes more money to them), and has no relation whatsoever with the employment or unemployment statistics. If you quit your job, take the uninsurance, and do not look for another job right away, then you have reduced the employment level, but you have not affected the unmployment rate.